= Solution
Fix $M_\star,\rho_p,t_\star$ and an initial-orbit ratio $\kappa=a_0/a_p$; use $\kappa=1$ for the figure. The qualitative <finite-age planetary scattering regimes> have two useful boundaries.
First, the order-unity escape-to-orbital-speed boundary $\chi=1$ is
$$
\boxed{M_{\rm esc}(a_p)=\sqrt{\frac{3}{32\pi}}\,
M_\star^{3/2}\rho_p^{-1/2}a_p^{-3/2}}.
$$
It slopes down with logarithmic slope $-3/2$. Above it a <planet> can give large fractional binding-energy changes, favouring <planetary ejection of a comet>; below it weak kicks and repeated collision opportunities favour <accretion>.
Second, equating the characteristic <comet energy diffusion> time to the age gives
$$
\boxed{M_{\rm age}(a_p)=M_\star
\sqrt{\frac{P_p}{100t_\star}}\;\kappa^{-1/4}
\ \propto\ M_\star^{3/4}a_p^{3/4}t_\star^{-1/2}\kappa^{-1/4}}.
$$
This boundary slopes up with logarithmic slope $3/4$. Above it the many-encounter diffusion estimate fits within the available age; below it that estimate exceeds the age.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/iii/2017-iii-paper-316-scattering-regimes.png]
{title=Regimes of planetary scattering and ejection}
{description=Qualitative <planetary scattering> map at fixed stellar <mass>, planetary <mass density>, age and $a_0=a_p$. The axes are normalized to the intersection of the $\chi=1$ and $t_{\rm ej}=t_\star$ boundaries.}
\b[Above both boundaries, rapid ejection is favoured]. Above the age boundary but below the escape-speed boundary, repeated encounters can act during the age while individual kicks remain weak, so collision or <accretion> is commonly favoured. Below the age boundary, the diffusion model predicts incomplete ejection; the escape-speed boundary still distinguishes strong from weak individual kicks.
These labels are statistical expectations under the stated encounter model. In particular, the given <diffusion coefficient> supplies no collision rate, so it does not prove that every object in the low-$\chi$ region is accreted within $t_\star$. Nor does a long ejection time exclude faster collisions or other loss mechanisms. Very massive <planets> with $N_{\rm ej}\lesssim1$ require a few-encounter treatment rather than an extrapolation of the diffusion formula.
Back to article page
